There exists variants of GW which are motivated by enriching the distance to account for additional information.
For instance many graph datasets [1]} contain vector features providing information on the nodes or the edges (such as atomic information for molecules represented as graphs).
The GW distance is not naturally able to handle these features, which motivated variants such as the Fused-GW distance [2]}, which penalizes features with OT and the graph structure with GW at the same time.
It is also possible to learn the nodes or edges' features as in [3]}.
By analogy with the low complexity of OT between univariate measures, there is a sliced version of Gromov-Wasserstein between Euclidean mm-spaces [4]}.
One can also focus on Euclidean spaces equipped with Gaussian distributions, but so far (in contrast with OT [5]}) the optimal plan is not known to be Gaussian.
Thus, assuming the transport plan is Gaussian, it is possible to derive a closed form which is an upper bound of \(\)  [6]}.
In the case of shapes it is also possible to replace the geodesic distance by the heat diffusion kernel [7]}.
